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A~$k$-associahedron is a simplicial complex whose facets, called~$k$-triangulations, are the inclusion maximal sets of diagonals of a convex polygon where no~$k+1$ diagonals mutually cross. Such complexes are conjectured for about a decade to have realizations as convex polytopes, and therefore as complete simplicial fans. Apart from four one-parameter families including simplices, cyclic polytopes and classical associahedra, only two instances of multiassociahedra have been geometrically realized so far. This paper reports on conjectural realizations for all~$2$-associahedra, obtained by heuristic methods arising from natural geometric intuition on subword complexes. Experiments certify that we obtain fan realizations of~$2$-associahedra of an~$n$-gon for~$n\in\{10,11,12,13\}$, further ones being out of our computational reach., 24 pages, 17 figures, 7 tables, source code added and reference to it in the paper

In this paper, we study the stability of catenoids and helicoids in the hyperbolic $3$-space $\mathbb{H}^3$. (1) For a family of spherical minimal catenoids $\{\mathcal{C}_a\}_{a>0}$ in $\mathbb{H}^3$, there exist two constants $0\bar{a}_c$., Comment: This paper replaces both arXiv:1204.4943 and arXiv:1502.04764

International audience; Let A * denote the free monoid generated by a finite nonempty set A. In this paper we introduce a new measure of complexity of languages L ⊆ A * defined in terms of the semigroup structure on A *. For each L ⊆ A * , we define its cost c(L) as the infimum of all real numbers α for which there exist a language S ⊆ A * with p S (n) = O(n α) and a positive integer k with L ⊆ S k. We also define the cost dimension d c (L) as the infimum of the set of all positive integers k such that L ⊆ S k for some language S with p S (n) = O(n c(L)). We are primarily interested in languages L given by the set of factors of an infinite word x = x 0 x 1 x 2 • • • ∈ A N of zero topological entropy, in which case c(L) < +∞. We establish the following characterisation of words of linear factor complexity: Let x ∈ A N and L = Fac(x) be the set of factors of x. Then p x (n) = Θ(n) if and only c(L) = 0 and d c (L) = 2. In other words, p x (n) = O(n) if and only if Fac(x) ⊆ S 2 for some language S ⊆ A + of bounded complexity (meaning lim sup p S (n) < +∞). In general the cost of a language L reflects deeply the underlying combinatorial structure induced by the semigroup structure on A *. For example, in contrast to the above characterisation of languages generated by words of sublinear complexity, there exist non factorial languages L of complexity p L (n) = O(log n) (and hence of cost equal to 0) and of cost dimension +∞. In this paper we investigate the cost and cost dimension of languages defined by infinite words of zero topological entropy. We establish the existence of words of cost zero and finite cost dimension having arbitrarily high polynomial complexity. In contrast we also show that for each α > 2 there exist infinite words x of positive cost and of complexity p x (n) = O(n α).

Strictly subadditive, subadditive and weakly subadditive labelings of quivers were introduced by the second author, generalizing Vinberg's definition for undirected graphs. In our previous work we have shown that quivers with strictly subadditive labelings are exactly the quivers exhibiting Zamolodchikov periodicity. In this paper, we classify all quivers with subadditive labelings. We conjecture them to exhibit a certain form of integrability, namely, as the $T$-system dynamics proceeds, the values at each vertex satisfy a linear recurrence. Conversely, we show that every quiver integrable in this sense is necessarily one of the $19$ items in our classification. For the quivers of type $\hat A \otimes A$ we express the coefficients of the recurrences in terms of the partition functions for domino tilings of a cylinder, called \emph{Goncharov-Kenyon Hamiltonians}. We also consider tropical $T$-systems of type $\hat A \otimes A$ and explain how affine slices exhibit solitonic behavior, i.e. soliton resolution and speed conservation. Throughout, we conjecture how the results in the paper are expected to generalize from $\hat A \otimes A$ to all other quivers in our classification., 46 pages, 21 figures; v2: all results concerning real rootedness of the recurrence polynomials have been restated as conjectures

Let E S ( n ) ES(n) be the smallest integer such that any set of E S ( n ) ES(n) points in the plane in general position contains n n points in convex position. In their seminal 1935 paper, Erdős and Szekeres showed that E S ( n ) ≤ ( 2 n − 4 n − 2 ) + 1 = 4 n − o ( n ) ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)} . In 1960, they showed that E S ( n ) ≥ 2 n − 2 + 1 ES(n) \geq 2^{n-2} + 1 and conjectured this to be optimal. In this paper, we nearly settle the Erdős-Szekeres conjecture by showing that E S ( n ) = 2 n + o ( n ) ES(n) =2^{n +o(n)} .

A knot in the 3-sphere is said to have zero negative unknotting number if it can be transformed into the unknot by performing only positive crossing changes. In this paper, we provide an obstruction for a knot to having zero negative unknotting number, and discuss its application to two classes of knots., 14 pages, 5 figures. This paper was published two years ago

Let $f\colon M\to N$ be a proper map between two aspherical compact orientable 3-manifolds with empty or toroidal boundary. We assume that $N$ is not a closed graph-manifold. Suppose that $f$ induces an epimorphism on fundamental groups. We show that $f$ is homotopic to a homeomorphism if one of the following holds: either for any finite-index subgroup $\Gamma$ of $\pi_1(N)$ the ranks of $\Gamma$ and of $f_*^{-1}(\Gamma)$ agree, or for any finite cover $\tilde{N}$ of $N$ the Heegaard genus of $\tilde{N}$ and the Heegaard genus of the pull-back cover $\tilde{M}$ agree., Comment: We split the old version into two papers. This version contains the results on epimorphisms of fundamental groups. The results on Grothendieck rigidity now appear in a separate paper

It has been 40 years since Lawson and Osserman introduced the three minimal cones associated with Dirichlet problems in their 1977 Acta paper [13] . The first cone was shown area-minimizing by Harvey and Lawson in the celebrated paper [10] . In this paper, we confirm that the other two are also area-minimizing. In fact, we show that every Lawson–Osserman cone of type ( n , p , 2 ) constructed in [26] is area-minimizing.

Inspired by work of McMullen, we show that any orbit of the diagonal group in the space of lattices accumulates on the set of stable lattices. As consequences, we settle a conjecture of Ramharter concerning the asymptotic behaviour of the Mordell constant, and reduce Minkowski's conjecture on products of linear forms to a geometric question, yielding two new proofs of the conjecture in dimensions up to 7., Comment: This paper was part of the paper "On stable lattices and the diagonal group" (arXiv:1309.4025) which was split into three shorter papers (now all on arXiv) following the referee's request

A recent paper of Shekhar compares the ranks of elliptic curves $E_1$ and $E_2$ for which there is an isomorphism $E_1[p] \simeq E_2[p]$ as $\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})$-modules, where $p$ is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where $p$ is a prime of good supersingular reduction., Comment: fixed a persistent typo in the results of Section 3. this typo led to the original example for p=3 being incorrect; it has been replaced with a correct example

This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $\lambda\left( G\right) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $k\geq1,$ $n\geq k^{3}/2+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $\delta\left( G\right) \geq k.$ If \[ \lambda\left( G\right) \geq n-k-1, \] then $G$ has a Hamiltonian cycle, unless $G=K_{1}\vee(K_{n-k-1}+K_{k})$ or $G=K_{k}\vee(K_{n-2k}+\overline{K}_{k})$. (2) Let $k\geq1,$ $n\geq k^{3}/2+k^{2}/2+k+5,$ and let $G$ be a graph of order $n$, with minimum degree $\delta\left( G\right) \geq k.$ If \[ \lambda\left( G\right) \geq n-k-2, \] then $G$ has a Hamiltonian path, unless $G=K_{k}\vee(K_{n-2k-1}+\overline {K}_{k+1})$ or $G=K_{n-k-1}+K_{k+1}$ In addition, it is shown that in the above statements, the bounds on $n$ are tight within an additive term not exceeding $2$., Comment: 18 pages. This version gives tighter bounds

We study the behavior of zero-divisors of the double zeta-function $\zeta_2(s_1,s_2)$. In our former paper \cite{MatSho14} we studied the case $s_1=s_2$, but in the present paper we consider the more general two variable situation. We carry out numerical computations in order to trace the behavior of zero-divisors. We observe that some divisors approach the points with $s_2=0$, while other divisors approach the points which are solutions of $\zeta(s_2)=1$., Comment: 14 pages, 16 figures

Let $R=K[[x_1,...,x_s]]$ be the ring of formal power series with maximal ideal $\mathfrak{m}$ over a field $K$ of arbitrary characteristic. On the ring $M_{m,n}$ of $m\times n$ matrices $A$ with entries in $R$ we consider several equivalence relations given by the action on $M_{m,n}$ of a group $G$. $G$ can be the group of automorphisms of $R$, combined with the multiplication of invertible matrices from the left, from the right, or from both sides, respectively. We call $A$ finitely $G$-determined if $A$ is $G$-equivalent to any matrix $B$ with ${A-B} \in {\mathfrak{m}^k M_{m,n}}$ for some finite integer $k$, which implies in particular that $A$ is $G$--equivalent to a matrix with polynomial entries. The classical criterion for analytic or differential map germs $f:(K^s,0) \to (K^m,0)$, $K = \mathbb{R}, \mathbb{C}$, says that $f \in M_{m,1}$ is finitely determined (with respect to various group actions) iff the tangent space to the orbit of $f$ has finite codimension in $M_{m,1}$. We extend this criterion to arbitrary matrices in $M_{m,n}$ if the characteristic of K is 0 or, more general, if the orbit map is separable. In positive characteristic however, the problem is more subtle since the orbit map is in general not separable, as we show by an example. This fact had been overlooked in previous papers. Our main result is a general sufficient criterion for finite $G$-determinacy in $M_{m,n}$ in arbitrary characteristic in terms of the tangent image of the orbit map, which we introduce in this paper. This criterion provides a computable bound for the $G$-determinacy of a matrix $A$ in $M_{m,n}$, which is new even in characteristic 0., Comment: 21 pages

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $\beta$-numbers. These $\beta$-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones' result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones' $\beta$ numbers. In this paper we give a version of P. Jones' theorem for sets of arbitrary (integer) dimension lying in Euclidean space. We estimate the $d$-dimensional Hausdorff measure of a set in terms of an analogous sum of $\beta$-type numbers. There is no assumption of Ahlfors regularity, but rather, only of a lower bound on the Hausdorff content. We adapt David and Semmes' version of Jones' $\beta$-numbers by redefining them using a Choquet integral. A key tool in the proof is G. David and T. Toro's parametrization of Reifenberg flat sets (with holes)., Comment: Corrected more typos. There are still several typos and small mistakes in the published version of the paper, so the authors will maintain an up-to-date version on their webpages as we continue to correct them

The main result (roughly) is that if (H_i) converges weakly to H and if also f(H_i) converges weakly to f(H), for a single strictly convex continuous function f, then (H_i) must converge strongly to H. One application is that if f(pr(H)) = pr(f(H)), where pr denotes compression to a closed subspace M, then M must be invariant for H. A consequence of this is the verification of a conjecture of Arveson, that Theorem 9.4 of [Arv] remains true in the infinite dimensional case. And there are two applications to operator algebras. If h and f(h) are both quasimultipliers, then h must be a multiplier. Also (still roughly stated) if h and f(h) are both in pA_sa p, for a closed projection p, then h must be strongly q-continuous on p., Comment: This paper is intended for a dual audience. Section 2, which contains the main result, is suitable for a general audience. Section 3 concerns technical operat or algebraic questions, though I have tried to present it with a minimum of tech nicality. As with all my papers, I welcome comments, especially for this paper, since I know almost nothing about Korovkin type theorems

The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. These are Nisnivich sheaves of $S^1$-spectra and the major computational tool of [GP1]. The aim of this paper is to show the following result which is essential in proving the main theorem of [GP1]: given an infinite perfect base field $k$, any $k$-smooth scheme $X$ and any $n\geq 1$, the map of simplicial pointed Nisnevich sheaves $(-,\mathbb{A}^1//\mathbb G_m)^{\wedge n}_+\to T^n$ induces a Nisnevich local level weak equivalence of $S^1$-spectra $$M_{fr}(X\times (\mathbb{A}^1// \mathbb G_m)^{\wedge n})\to M_{fr}(X\times T^n).$$ Moreover, it is proven that the sequence of $S^1$-spectra $$M_{fr}(X \times T^n \times \mathbb G_m) \to M_{fr}(X \times T^n \times\mathbb A^1) \to M_{fr}(X \times T^{n+1})$$ is locally a homotopy cofiber sequence in the Nisnevich topology. Another important result of this paper shows that homology of framed motives is computed as linear framed motives in the sense of [GP1]. This computation is crucial for the whole machinery of framed motives [GP1]., Comment: This is the final revised version

Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let Y denote the set of fixed points of H in X, and N(H) the normalizer of H in G. In this paper we study the natural map from the quotient of Y by N(H) to the quotient of X by G induced by the inclusion of Y in X. We show that, given G and H, this map is a finite morphism for all G-varieties X if and only if H is G-completely reducible (in the sense defined by J-P. Serre); this was proved in characteristic zero by Luna in the 1970s. We discuss some applications and give a criterion for the map of quotients to be an isomorphism. We show how to extend some other results in Luna's paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; then the double coset HgK is closed for generic g in G if and only if the intersection of generic conjugates of H and K is reductive., Comment: 29 pages; some changes in Section 5; to appear in Math. Z

The purpose of this paper is to modify the notion of the Nevanlinna constant $\operatorname{Nev}(D)$, recently introduced by the first author, for an effective Cartier divisor on a projective variety $X$. The modified notion is called the birational Nevanlinna constant and is denoted by $\operatorname{Nev}_{\text{bir}}(D)$. By computing $\operatorname{Nev}_{\text{bir}}(D)$ using the filtration constructed by Autissier in 2011, we establish a general result (see the General Theorem in the Introduction), in both the arithmetic and complex cases, which extends to general divisors the 2008 results of Evertse and Ferretti and the 2009 results of the first author. The notion $\operatorname{Nev}_{\text{bir}}(D)$ is originally defined in terms of Weil functions for use in applications, and it is proved later in this paper that it can be defined in terms of local effectivity of Cartier divisors after taking a proper birational lifting. In the last two sections, we use the notion $\operatorname{Nev}_{\text{bir}}(D)$ to recover the proof of an example of Faltings from his 2002 Baker's Garden article., Comment: 56 pages; latex

The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In 2015 we gave two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. In this paper we give an elementary proof of Naruse's formula based on the case of border strips. For special border strips, we obtain curious new formulas for the Euler and $q$-Euler numbers in terms of certain Dyck path summations., Comment: 33 pages, 10 figures. This is the second paper of the series "Hook formulas for skew shapes". Most of Sections 8 and 9 in this paper used to be part of arxiv:1512.08348 (v1,v2); v2 fixed several typos; v3 made precision in definition of flagged tableaux in Section 3.2 and fixed typo in Example 3.3; v4 fixed small typos in proof of Corollary 7.6

We develop a new approach to random walks on de Bruijn graphs over the alphabet $A$ through right congruences on $A^k$, defined using the natural right action of $A^+$. A major role is played by special right congruences, which correspond to semaphore codes and allow an easier computation of the hitting time. We show how right congruences can be approximated by special right congruences., Comment: 34 pages; 10 figures; as requested by the journal, the previous version of this paper was divided into two; this version contains Sections 1-8 of version 1; Sections 9-12 will appear as a separate paper with extra material added

This paper concerns the worst-case complexity of cyclic coordinate descent (C-CD) for minimizing a convex quadratic function, which is equivalent to Gauss-Seidel method and can be transformed to Kaczmarz method and projection onto convex sets (POCS). We observe that the known provable complexity of C-CD can be $O(n^2)$ times slower than randomized coordinate descent (R-CD), but no example was rigorously proven to exhibit such a large gap. In this paper we show that the gap indeed exists. We prove that there exists an example for which C-CD takes at least $O(n^4 \kappa_{\text{CD}} \log\frac{1}{\epsilon})$ operations, where $\kappa_{\text{CD}}$ is related to Demmel's condition number and it determines the convergence rate of R-CD. It implies that in the worst case C-CD can indeed be $O(n^2)$ times slower than R-CD, which has complexity $O( n^2 \kappa_{\text{CD}} \log\frac{1}{\epsilon})$. Note that for this example, the gap exists for any fixed update order, not just a particular order. Based on the example, we establish several almost tight complexity bounds of C-CD for quadratic problems. One difficulty with the analysis is that the spectral radius of a non-symmetric iteration matrix does not necessarily constitute a \textit{lower bound} for the convergence rate. An immediate consequence is that for Gauss-Seidel method, Kaczmarz method and POCS, there is also an $O(n^2) $ gap between the cyclic versions and randomized versions (for solving linear systems). We also show that the classical convergence rate of POCS by Smith, Solmon and Wager [1] is always worse and sometimes can be infinitely times worse than our bound., Comment: 47 pages. Add a few tables to summarize the main convergence rates; add comparison with classical POCS bound; add discussions on another example

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k, 0 or 1 , by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m=5,7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n2m+1, m as sum of row terms of corresponding triangle. This version might be out of date, proceed by the link to see actual version., 16 pages, 8 figures, 2 tables, typos revised, added missing references, results generalized and shortened, derivations detailed.

The article A “regular” pentagonal tiling of the plane by P. L. Bowers and K. Stephenson, Conform.Geom. Dyn. 1, 58–86, 1997, defines a conformal pentagonal tiling. This is a tiling of the planewith remarkable combinatorial and geometric properties. However, it doesn’t have finite localcomplexity in any usual sense, and therefore we cannot study it with the usual tiling theory. Theappeal of the tiling is that all the tiles are conformally regular pentagons. But conformal mapsare not allowable under finite local complexity. On the other hand, the tiling can be describedcompletely by its combinatorial data, which rather automatically has finite local complexity. Inthis paper we give a construction of the discrete hull just from the combinatorial data. The mainresult of this paper is that the discrete hull is a Cantor space.

Let $G$ be a finite group, and let $\kappa(G)$ be the probability that elements $g$, $h\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $\kappa(G) \geq 1/4$, and shows that $G$ is abelian whenever $\kappa(G)|G| < 7/4$. It is also shown that $\kappa(G)|G|$ depends only on the isoclinism class of $G$. Specialising to the symmetric group $S_n$, the paper shows that $\kappa(S_n) \leq C/n^2$ for an explicitly determined constant $C$. This bound leads to an elementary proof of a result of Flajolet \emph{et al}, that $\kappa(S_n) \sim A/n^2$ as $n\rightarrow \infty$ for some constant $A$. The same techniques provide analogous results for $\rho(S_n)$, the probability that two elements of the symmetric group have conjugates that commute., Comment: 34 pages, corrected version, to appear in Journal of the London Mathematical Society

We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type., 27 pages, no figures, submitted, Israel J. Math. It turns out that there is one specific family of algebraic group - namely Sp(4) in characteristic 3 - which our methods are unable to resolve, and we have adjusted the paper to exclude this case from the results

We confirm, to some extent, the belief that a projective variety X has the largest number (relative to the dimension of X) of independent commuting automorphisms of positive entropy only when X is birational to a complex torus or a quotient of a torus. We also include an addendum to an early paper though it is not used in the present paper., Comment: Mathematische Annalen (to appear)

In 1998, Kleinbock & Margulis established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_{0}(\vv x) = 1/n$ for almost every point $\vv x$ on a non-degenerate submanifold $\cM$ of $\R^n$. In this paper the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any `inhomogeneous' vector $\bm\theta\in\R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_{0}(\vv x, \bm\theta)= 1/n$ for almost every point $\vv x$ on $M$. The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent $w_0(\vv x)=1/n$ for almost all $\vv x\in \cM$ if and only if for any $\bm\theta\in\R^n$ the inhomogeneous exponent $w_0(\vv x,\bm\theta)=1/n$ for almost all $\vv x\in \cM$. The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered in \cite{Beresnevich-Velani-new-inhom}. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas of \cite{Beresnevich-Velani-new-inhom} while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory., Comment: Dedicated to A.O. Gelfond on what would have been his 100th birthday 13 pages

In this paper we get an estimate of Favard length of an arbitrary neighbourhood of an arbitrary self-similar Cantor set. Consider $L$ closed disjoint discs of radius $1/L$ inside the unit disc. By using linear maps of smaller disc onto the unit disc we can generate a self-similar Cantor set $G$. Then $\G=\bigcap_n\G_n$. One may then ask the rate at which the Favard length - the average over all directions of the length of the orthogonal projection onto a line in that direction - of these sets $\G_n$ decays to zero as a function of $n$. The quantitative results for the Favard length problem were obtained by Peres-Solomyak and Tao; in the latter paper a general way of making a quantitative statement from the Besicovitch theorem is considered. But being rather general, this method does not give a good estimate for self-similar structures such as $\G_n$. Indeed, vastly improved estimates have been proven in these cases: in the paper of Nazarov-Peres-Volberg, it was shown that for 1/4 corner Cantor set one has $p0$. In the present work we give an estimate that works for {\it any} Besicovitch set which is self-similar. However estimate is worse than the power one. The power estimate still appears to be related to a certain regularity property of zeros of a corresponding linear combination of exponents (we call this property {\it analytic tiling})., several misprints were corrected, 36 pages

In this paper, we study the quotient and “blow-up” of graph-of-groups [Formula: see text] and of their automorphisms [Formula: see text]. We show that the existence of such a blow-up of any [Formula: see text], relative to a given family of “local” graph-of-groups isomorphisms [Formula: see text] depends crucially on the [Formula: see text]-conjugacy class of the correction term [Formula: see text] for any edge [Formula: see text] of [Formula: see text], where [Formula: see text]-conjugacy is a new but natural concept introduced here. As an application, we obtain a criterion as to whether a partial Dehn twist can be blown up relative to local Dehn twists, to give an actual Dehn twist. The results of this paper are also used crucially in the follow-up papers [Lustig and Ye, Normal form and parabolic dynamics for quadratically growing automorphisms of free groups, arXiv:1705.04110v2; Ye, Partial Dehn twists of free groups relative to local Dehn twists — A dichotomy, arXiv:1605.04479 ; When is a polynomially growing automorphism of [Formula: see text] geometric, arXiv:1605.07390 ].

In this paper, we study some aspects of the irreducibility of $\widetilde{M_g^{Pl}(G)}$ and its interrelation with the existence of "normal forms", i.e. non-singular plane equations (depending on a set of parameters) such that a specialization of the parameters gives a certain non-singular plane model associated to the elements of $\widetilde{M_g^{Pl}(G)}$. In particular, we introduce the concept of being equation strongly irreducible (ES-Irreducible) for which the locus $\widetilde{M_g^{Pl}(G)}$ is represented by a single "normal form". Henn, and Komiya-Kuribayashi, observed that $\widetilde{M_3^{Pl}(G)}$ is ES-Irreducible. In this paper we prove that this phenomena does not occur for any odd $d>4$. More precisely, let $\mathbb{Z}/m\mathbb{Z}$ be the cyclic group of order $m$, we prove that $\widetilde{M_g^{Pl}(\mathbb{Z}/(d-1)\mathbb{Z})}$ is not ES-Irreducible for any odd integer $d\geq5$, and the number of its irreducible components is at least two. Furthermore, we conclude the previous result when $d=6$ for the locus $\widetilde{M_{10}^{Pl}(\mathbb{Z}/3\mathbb{Z})}$. Lastly, we prove the analogy of these statements when $K$ is any algebraically closed field of positive characteristic $p$ such that $p>(d-1)(d-2)+1$., This paper is a recent version of chapter 1 of arXiv:1503.01149

As we proved earlier, for a triangulated category $\underline{C}$ endowed with a weight structure $w$ and a triangulated subcategory $\underline{D}$ of $\underline{C}$ (strongly) generated by cones of a set of morphisms $S$ in the heart $\underline{Hw}$ of $w$ there exists a weight structure $w'$ on the Verdier quotient $\underline{C}'=\underline{C}/\underline{D}$ such that the localization functor $\underline{C} \to \underline{C}'$ is weight-exact (i.e., "respects weights"). The goal of this paper is to find conditions ensuring that for any object of $\underline{C}'$ of non-negative (resp. non-positive) weights there exists its preimage in $\underline{C}$ satisfying the same condition; we call a certain stronger version of the latter assumption the left (resp., right) weight lifting property. We prove that these weight lifting properties are fulfilled whenever the set $S$ satisfies the corresponding (left or right) Ore conditions. Moreover, if $\underline{D}$ is generated by objects of $\underline{Hw}$ then any object of $\underline{Hw}'$ lifts to $\underline{Hw}$. We apply these results to obtain some new results on Tate motives and finite spectra (in the stable homotopy category). Our results are also applied to the study of the so-called Chow-weight homology in another paper., v3: 25 pages, A full revision has been made, an author added

We develop Jacobson’s refinement of Engel’s Theorem for Leibniz algebras. We then note some consequences of the result. Since Leibniz algebras were introduced in [Loday 1993] as a noncommutative generalization of Lie algebras, one theme has been to extend Lie algebra results to Leibniz algebras. In particular, Engel’s theorem has been extended in [Ayupov and Omirov 1998; Barnes 2011; Patsourakos 2007]. In the second of these works, the classical Engel’s theorem is used to give a short proof of the result for Leibniz algebras. The proofs in the other two papers do not use the classical theorem and, therefore, the Lie algebra result is included in the result. In this note, we give two proofs of the generalization to Leibniz algebras of Jacobson’s refinement to Engel’s theorem, a short proof which uses Jacobson’s theorem and a second proof which does not use it. It is interesting to note that the technique of reducing the problem to the special Lie algebra case significantly shortens the proof for the general Leibniz algebras case. This approach has been used in a number of situations [Barnes 2011]. We also note some standard consequences of this theorem. The proofs of the corollaries are exactly as in Lie algebras (see [Kaplansky 1971]). Our result can be used to directly show that the sum of nilpotent ideals is nilpotent, and hence one has a nilpotent radical. In this paper, we consider only finite dimensional algebras and modules over a field F. An algebra A is called Leibniz if it satisfies x.yz/D.x y/zC y.x z/. Denote by Ra and L a, respectively, right and left multiplication by a2 A. Then RbcD Rc RbC L b Rc

Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. In this paper, we introduce the atomic Hardy space $H^1(\mu)$ and prove that its dual space is the known space ${\rm RBMO}(\mu)$ in this context. Using this duality, we establish a criterion for the boundedness of linear operators from $H^1(\mu)$ to any Banach space. As an application of this criterion, we obtain the boundedness of Calder\'on--Zygmund operators from $H^1(\mu)$ to $L^1(\mu)$., Comment: This paper has been withdrawn by the authors, since it has already been published

In this paper we construct explicitly a square integrable residual automorphic representation of the special orthogonal group $SO_{2n}$, through Eisenstein series. We show that this representation comes from an elliptic Arthur parameter $\psi$ and appears in the space $L^2(SO_{2n}(\mathbb{Q})\backslash SO_{2n}(\mathbb{A}_{\mathbb{Q}}))$ with multiplicity one. Next, we consider parameters whose Hecke matrices, at the unramified places, have eigenvalues bigger (in absolute value), than those of the parameter constructed before. The main result is, that these parameters cannot be cuspidal. We establish bounds for the eigenvalues of Hecke operators, as consequences of Arthur's conjectures for $SO_{2n}$. Next, we calculate the character and the twisted characters for the representations that we constructed. Finally, we find the composition of the global and local Arthur's packets associated to our parameter $\psi$. All the results in this paper are true if we replace $\mathbb{Q}$ by any number field $F$., Comment: 39 pages

As a first goal, it is explained why Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of P^m only for m < 16. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential, but critical, high-order obstructions in the corresponding Taylor towers. For m > 15, the relation TC^S(P^m) > n-1 is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an n-dimensional Euclidean embedding of P^m. A speculative bordism-type form of this primary obstruction is proposed as an analogue of Davis' BP-approach to the immersion problem of P^m. A form of the Euler class viewpoint is applied to show TC^S(P^3) = 5, as well as to suggest a few higher dimensional projective spaces for which the method could produce new information. As a second goal, the paper extends Farber's work on the motion planning problem in order to develop the notion of a symmetric motion planner for a mechanical system S. Following Farber's lead, this concept is connected to the symmetric topological complexity of the state space of S. The paper ends by sketching the construction of a concrete 5-local-rules symmetric motion planner for P^3., updated, final version to appear in Transactions of the American Mathematical Society

The purpose of this announcement is to describe a development given in a series of forthcoming papers by the authors that concern operators of the form \[ f\mapsto \psi(x) \int f(\gamma_t(x)) K(t)\: dt, \] where $\gamma_t(x)=\gamma(t,x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \mathbb{R}^N\times \mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $K(t)$ is a "multi-parameter singular kernel" supported near $t=0$, and $\psi$ is a cutoff function supported near $x=0$. This note concerns the case when $K$ is a "product kernel". The goal is to give conditions on $\gamma$ such that the above operator is bounded on $L^p$ for $1, Comment: 18 pages; an announcement for a three part series of papers; final version to appear in Math. Res. Lett

Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]\times [a_2,b_2]\times\cdots\times [a_k,b_k]$. The boxicity of $G$, box$(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional boxes; i.e., each vertex is mapped to a $k$-dimensional box and two vertices are adjacent in $G$ if and only if their corresponding boxes intersect. Let $\mathcal{P}=(S,P)$ be a poset, where $S$ is the ground set and $P$ is a reflexive, antisymmetric and transitive binary relation on $S$. The dimension of $\mathcal{P}$, $\dim(\mathcal{P})$, is the minimum integer $t$ such that $P$ can be expressed as the intersection of $t$ total orders. Let $G_{\mathcal{P}}$ be the underlying comparability graph of $\mathcal{P}$; i.e., $S$ is the vertex set and two vertices are adjacent if and only if they are comparable in $\mathcal{P}$. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset $\mathcal{P}$, box$(G_{\mathcal{P}})/(\chi(G_{\mathcal{P}})-1) \le \dim(\mathcal{P})\le 2\mbox{box}(G_{\mathcal{P}})$, where $\chi(G_{\mathcal{P}})$ is the chromatic number of $G_{\mathcal{P}}$ and $\chi(G_{\mathcal{P}})\ne1$. It immediately follows that if $\mathcal{P}$ is a height-$2$ poset, then box$(G_{\mathcal{P}})\le \dim(\mathcal{P})\le 2\mbox{box}(G_{\mathcal{P}})$ since the underlying comparability graph of a height-$2$ poset is a bipartite graph. The second result of the paper relates the boxicity of a graph $G$ with a natural partial order associated with the extended double cover of $G$, denoted as $G_c$: Note that $G_c$ is a bipartite graph with partite sets $A$ and $B$ which are copies of $V(G)$ such that, corresponding to every $u\in V(G)$, there are two vertices $u_A\in A$ and $u_B\in B$ and $\{u_A,v_B\}$ is an edge in $G_c$ if and only if either $u=v$ or $u$ is adjacent to $v$ in $G$. Let $\mathcal{P}_c$ be the natural height-$2$ poset associated with $G_c$ by making $A$ the set of minimal elements and $B$ the set of maximal elements. We show that $\frac{\mbox{box}(G)}{2} \le \dim(\mathcal{P}_c) \le 2\mbox{box}(G)+4$. These results have some immediate and significant consequences. The upper bound $\dim(\mathcal{P})\le 2\mbox{box}(G_\mathcal{P})$ allows us to derive hitherto unknown upper bounds for poset dimension such as $\dim(\mathcal{P})\le 2\mbox{ tree width }(G_{\mathcal{P}})+4$, since boxicity of any graph is known to be at most its tree width $+\; 2$. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree $\Delta$ is $O(\Delta\log^2\Delta)$, which is an improvement over the best-known upper bound of $\Delta^2+2$. (2) There exist graphs with boxicity $\Omega(\Delta\log\Delta)$. This disproves a conjecture that the boxicity of a graph is $O(\Delta)$. (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on $n$ vertices with a factor of $O(n^{0.5-\epsilon})$ for any $\epsilon>0$ unless $NP=ZPP$.

Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for deriving all packings of n spheres in R^3 satisfying minimal rigidity constraints (>= 3 contacts per sphere and >= 3n-6 total contacts). We derive such packings for n = 9, (iii) the number of ground states (i.e. packings with the maximal number of contacts) oscillates with respect to n, (iv) for 10, The updates that have been made to version II are as follows: (i) some updates to the main text have been made, primarily correcting the number of packings reported in PRL, 103, 118303 and in version I of this paper for n = 9,10. (ii) The supplemental information for the paper has been included

In his paper "Finite groups have many conjugacy classes" (J. London Math. Soc (2) 46 (1992), 239-249), L. Pyber proved the to date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber's paper., 13 pages

In this paper we prove the birational superrigidity of Fano-Mori fibre spaces $\pi\colon V\to S$, every fibre of which is a complete intersection of type $d_1\cdot d_2$ in the projective space ${\mathbb P}^{d_1+d_2}$, satisfying certain conditions of general position, under the assumption that the fibration $V/S$ is sufficiently twisted over the base (in particular, under the assumption that the $K$-condition holds). The condition of general position for every fibre guarantees that the global log canonical threshold is equal to one. This condition bounds the dimension of the base $S$ by a constant that depends on the dimension $M$ of the fibre only (as the dimension $M$ of the fibre grows, this constant grows as $\frac12 M^2$). The fibres and the variety $V$ itself may have quadratic and bi-quadratic singularities, the rank of which is bounded from below., Comment: 86 pages, the final version

We determine the order of magnitude of H^{(k+1)}(x,\vec{y},2\vec{y}), the number of integers up to x that are divisible by a product d_1...d_k with y_i, Comment: 34 pages. Added reference [10] to a paper of Nair and Tenenbaum which contains a result similar to Lemma 2.2 and which appeared prior to the publication of this paper. Simplified the proof of Lemma 2.2 using ideas from [10]. Removed part (a) of Lemma 2.2, as it is now redundant. Version 5 remains the published version

Let $a,b,c $ be any positive integers such that $c\mid ab$ and $d_i^\pm$ is a square free positive integer of the form $d_i^\pm=a^{2k} b^{2l}\pm i c^m$ where $k,l \geq m$ and $i=1,2.$ The main focus of this paper to find the fundamental solution of the equation $ x^2-d_i^\pm y^2=1$ with the help of the continued fraction of $\sqrt{d_i^\pm}.$ We also obtain all the positive solutions of the equations $ x^2-d_i^\pm y^2=\pm 1$ and $ x^2-d_i^\pm y^2=\pm 4$ by means of the Fibonacci and Lucas sequences. Furthermore, in this work, we derive some algebraic relations on the Pell form $ F_{d_i^\pm}(x, y) = x^2-d_i^\pm y^2 $ including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation $ F_{\Delta_{d_i^\pm}} (x, y) = 1 $ in terms of $d_i^\pm. We generalized all the results of the papers [2], [9], [26], and [37]., Comment: 16 pages

We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| = ��then there is a bounded complexity 3-step nilsequence F(g(n)��) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s >= 4 as well, and a longer paper will follow concerning this. By combining this with several previous papers of the first two authors one obtains the generalised Hardy-Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p_1 < p_2 < p_3 < p_4 < p_5, 49 pages, to appear in Glasgow J. Math. Fixed a problem with the file (the paper appeared in duplicate)

We present a new generalization of the classical trisecant lemma. Our approach is quite different from previous generalizations. Let $X$ be an equidimensional projective variety of dimension $d$. For a given $k \leq d + 1$, we are interested in the study of the variety of $k$-secants. The classical trisecant lemma just considers the case where $k = 3$ while elsewhere the case $k = d + 2$ is considered. Secants of order from $4$ to $d + 1$ provide service for our main result. In this paper, we prove that if the variety of $k$-secants ($k \leq d + 1$) satisfies the three following conditions: (i) trough every point in $X$, passes at least one $k$-secant, (ii) the variety of $k$-secant satisfies a strong connectivity property that we defined in the sequel, (iii) every $k$-secant is also a ($k+1$)-secant, then the variety $X$ can be embedded into $P^{d+1}$. The new assumption, introduced here, that we called strong connectivity is essential because a naive generalization that does not incorporate this assumption fails as we show in some example. The paper concludes with some conjectures concerning the essence of the strong connectivity assumption., Comment: arXiv admin note: text overlap with arXiv:0712.3878

Let $\Cant_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $\K_n = \Cant_n \times \Cant_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $\K_n$ is essentially the average length of the projections of $\K_n$, also known as the Favard length of $\K_n$. A classical theorem of Besicovitch implies that the Favard length of $\K_n$ tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was $\exp(- c\log_* n)$, due to Peres and Solomyak. ($\log_* n$ is the number of times one needs to take log to obtain a number less than 1 starting from $n$). In Nazarov-Peres-Volberg paper (arxiv:math 0801.2942) the power estimate from above was obtained. The exponent in this paper was less than 1/6 but could have been slightly improved. On the other hand, a simple estimate shows that from below we have the estimate $\frac{c}{n}$. Here we apply the idea from papers of Nets Katz (MRL (1996), 527-536) and Bateman-Katz (arxiv:math/0609187v1 2006) to show that the estimate from below can be in fact improved to $c \frac{\log n}{n}$. This is in drastic difference from the case of {\em random} Cantor sets studied by Peres and Solomyak in Pacific J. Math. 204 (2002), 473-496., Comment: 11 pages, one figure

A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal of this paper is to show the abundance of elements of N_p. Our main result shows that any divisor n of q-1, where q is a power of p, such that $n\ge (p-1)^{1/p} (q-1)^{1-1/(2p)}$, belongs to N_p. This extends its special case for p=2 which was proved in a previous paper by a different method., 10 pages. This version has been revised according to a referee's suggestions. The additions include a discussion of the (lower) density of the set N_p, and the results of more extensive machine computations. Note that the title has also changed. To appear in Israel J. Math

In this paper, a random graph process ${G(t)}_{t\geq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{t\geq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on $G(t-1)$, the probability that a given edge is connected to vertex i is proportional to $d_i(t-1)+\delta$, where $d_i(t-1)$ is the degree of vertex $i$ at time $t-1$, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent $\tau=\min\{\tau_{W}, \tau_{P}\}$, where $\tau_{W}$ is the power-law exponent of the initial degrees $(W_t)_{t\geq 1}$ and $\tau_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed., Comment: In the published form of the paper, the proof of Proposition 2.1 is incomplete. This version contains the complete proof

Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60], the author claimed some surprising connections among these seemingly unrelated fascinating areas. In this paper we establish further connections between zero-sum problems for abelian p-groups and covers of the integers. For example, we extend the famous Erdos-Ginzburg-Ziv theorem in the following way: If {a_s(mod n_s)}_{s=1}^k covers each integer either exactly 2q-1 times or exactly 2q times where q is a prime power, then for any c_1,...,c_k in Z/qZ there exists a subset I of {1,...,k} such that sum_{s in I}1/n_s=q and sum_{s in I}c_s=0. Our main theorem in this paper unifies many results in the two realms and also implies an extension of the Alon-Friedland-Kalai result on regular subgraphs.

Given a finite poset $\mathcal P$, we call a family $\mathcal F$ of subsets of $[n]$ $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturated number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we are mainly interested in the four-point poset called the butterfly. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan showed that the saturation number for the butterfly lies between $\log_2{n}$ and $n^2$. We give a linear lower bound of $n+1$. We also prove some other results about the butterfly and the poset $\mathcal N$., Comment: 13 pages

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator $$\overline{\partial }$$ ∂ ¯ on spaces $${\mathcal {E}}{\mathcal {V}}(\varOmega ,E)$$ E V ( Ω , E ) of $${\mathcal {C}}^{\infty }$$ C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights $${\mathcal {V}}$$ V . Vector-valued means that these functions have values in a locally convex Hausdorff space E over $${\mathbb {C}}$$ C . We derive a counterpart of the Grothendieck-Köthe-Silva duality $${\mathcal {O}}({\mathbb {C}}\setminus K)/{\mathcal {O}}({\mathbb {C}})\cong {\mathscr {A}}(K)$$ O ( C \ K ) / O ( C ) ≅ A ( K ) with non-empty compact $$K\subset {\mathbb {R}}$$ K ⊂ R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of $$\overline{\partial }:{\mathcal {E}} {\mathcal {V}}(\varOmega ,E)\rightarrow {\mathcal {E}}{\mathcal {V}} (\varOmega ,E)$$ ∂ ¯ : E V ( Ω , E ) → E V ( Ω , E ) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $${\mathcal {E}}{\mathcal {V}}(\varOmega ,{\mathbb {C}})$$ E V ( Ω , C ) .